Model Examination paper-I

1(i) Let be a probability space. Prove or disprove the following. If and

, then . [5]

1(ii) Define a filed and give an example of a field which is not a -field. [5]

2(i) Let be a random variable. Is is independent of iff is a constant random variable.

[5]

2(ii) Check whether the following function is a distribution function.

[5]

3(i) Let be a normal random variable with mean and variance . Find such that

is standard normal. [5]

3(ii) Let and be independent and identically distributed geometric random variables with

parameter . Find and . [5]

4(i) Let be a Poisson random variable with mean . Show that

[5]

4(ii) Let be continuous random variable with joint pdfs

Find the marginal pdfs of . [5]

5(i) Let be a sequence of random variables. Show that in distribution

iff in probability. [7]

5(ii) State central limit theorem. [3]